Symmetry, Regulator Dependence, and Functional-RG Error Control

A functional regulator changes the quadratic kernel and can therefore preserve, deform, or break a symmetry at finite scale. The correct test is not always the unmodified Ward identity: one must derive the identity obeyed by the regulated functional, solve it consistently with the flow, and check that the physical identity is recovered when the regulator is removed. A finite truncation adds separate ansatz, projection, closure, and numerical errors.

This page derives the modified identity for a linear global symmetry, specializes it to an $O(N)$ scalar flow, explains the modified Slavnov–Taylor and splitting identities needed in gauge and background-field calculations, and gives convexity and error-budget reliability criteria. Regulator variation is one diagnostic among several; it is never promoted by itself to an uncertainty theorem.

Required background. Functional-RG Truncations and Projection Methods defines ansatz, projector, closure, nested enlargement, and the validation checklist reused below.

Helpful background. Coupling to Background Gauge Fields and Bundles supplies background-field transformations. Slavnov–Taylor and Zinn-Justin Identities supplies BRST sources and the unregulated master identity.

Ward identities with an infrared kernel

Collect real bosonic fields into $\chi$ and consider an infinitesimal linear transformation

$$\delta\chi=\epsilon T\chi.$$

Assume that the microscopic action and measure are invariant, but add

$$\Delta S_k[\chi] = \frac12\chi\cdot R_k\cdot\chi.$$

Its variation is

$$\delta\Delta S_k = \frac{\epsilon}{2}\, \chi\cdot K_{T,k}\cdot\chi, \qquad K_{T,k} \equiv T^{\mathsf T}R_k+R_kT.$$

A change of integration variable in the regulated source functional gives

$$J\cdot T\varphi = \frac12\varphi\cdot K_{T,k}\cdot\varphi + \frac12\operatorname{Tr}(K_{T,k}G_k),$$

where $G_k=(\Gamma_k^{(2)}+R_k)^{-1}$. Using $J=\Gamma_k^{(1)}+R_k\varphi$, the field-quadratic terms cancel and leave the modified Ward identity

$$\boxed{ \mathcal W_T\Gamma_k \equiv \Gamma_k^{(1)}\cdot T\varphi = \frac12\operatorname{Tr} \left[ G_k \left(T^{\mathsf T}R_k+R_kT\right) \right] }.$$

This equation separates three cases.

Regulator relation to the symmetry	Finite-$k$ identity	Required endpoint statement
$T^{\mathsf T}R_k+R_kT=0$	The ordinary Ward identity holds exactly.	Preserve it along the flow and in the projection.
The regulator breaks the symmetry in a known way	The trace on the right is a calculable insertion.	Show that the insertion and the identity residual vanish as $R_k\to0$.
The truncation cannot represent the required insertion	The projected flow and identity are not closed together.	Enlarge the ansatz or restrict the physical claim.

The right-hand side is not a free symmetry-breaking counterterm. It is fixed by the same regulator and full regularized propagator that appear in the flow. Evaluating the flow with one propagator approximation and the identity with another invalidates the consistency test.

An O(N) scalar monitor

For an $O(N)$ vector, take antisymmetric generators $T^{ab}$ and an internal-space covariant kernel

$$R_k^{ab}(q)=\delta^{ab}R_k(q).$$

Then $T^{\mathsf T}R_k+R_kT=0$: this regulator does not break $O(N)$. An invariant derivative ansatz written in terms of $\rho=\varphi^a\varphi^a/2$ obeys

$$\mathcal W^{ab}\Gamma_k = \int d^dx\, \left( \varphi^a\frac{\delta\Gamma_k}{\delta\varphi^b} - \varphi^b\frac{\delta\Gamma_k}{\delta\varphi^a} \right) =0.$$

At a constant background $\bar\varphi=(\sqrt{2\rho},0,\ldots,0)$, the same identity relates the equation of state to the transverse inverse propagator:

$$\mathcal E_W(k,\rho) \equiv \Gamma_{k,T}^{(2)}(p=0;\rho) - \frac{\Gamma_{k,L}^{(1)}(\rho)}{\sqrt{2\rho}} =0.$$

For $\Gamma_k=\int[\,\frac12(\partial\varphi)^2+U_k(\rho)\,]$, both terms equal $U_k'(\rho)$. At a stationary broken minimum, $U_k'(\kappa_k)=0$ and the $N-1$ transverse modes become massless as the regulator is removed. On a grid, monitor an absolute residual near $\rho=0$ and a normalized residual elsewhere; a relative residual alone becomes meaningless when both sides vanish.

An ansatz built entirely from $O(N)$ invariants can make $\mathcal E_W=0$ algebraically. That is a valuable protection against symmetry-breaking numerics, but it does not test omitted momentum dependence. A vertex truncation that projects longitudinal and transverse sectors separately supplies a more independent check.

Two regulator profiles at one loop

The local-potential flow is

$$\begin{aligned} \partial_tU_k(\rho) = \frac12\int_q\partial_tR_k(q) \bigg[ & \frac{N-1}{q^2+R_k+U_k'} \\ &+ \frac{1}{q^2+R_k+U_k'+2\rho U_k''} \bigg]. \end{aligned}$$

Use $U_k=m_k^2\rho+\lambda_k\rho^2/6+\cdots$ and expand at $m_k=0$. In four dimensions,

$$\partial_t\lambda = \frac{N+8}{3}\lambda^2\,\mathcal I_R + O(\lambda^3), \qquad \mathcal I_R \equiv \int_q \frac{\partial_tR_k(q)} {\left[q^2+R_k(q)\right]^3}.$$

Write $R_k(q)=q^2r(y)$ with $y=q^2/k^2$, fixed field normalization, $r(0)=\infty$, and $r(\infty)=0$. Radial integration turns the threshold integral into a boundary term:

$$\mathcal I_R = \frac{1}{16\pi^2} \left[ \frac{1}{(1+r(y))^2} \right]_{y=0}^{y=\infty} = \frac{1}{16\pi^2}.$$

Therefore both

$$r_{\mathrm{opt}}(y) = \left(\frac1y-1\right)\theta(1-y), \qquad r_{\exp}(y) = \frac1{e^y-1},$$

give

$$\partial_t\lambda = \frac{N+8}{48\pi^2}\lambda^2 + O(\lambda^3).$$

This simultaneous zero Ward residual and zero one-loop regulator spread checks group factors, normalization, and endpoint conditions. It does not predict zero regulator spread for nonperturbative observables in a finite truncation. Exact large-$N$ flows further show that universal quantities can be regulator independent while coupling coordinates remain regulator dependent, so the scan should target physical or universal outputs. Knorr 2021, preprint §§ II.F, III.D, and IV, pp. 9, 13–14

Modified Slavnov–Taylor identities

Gauge fixing replaces naive gauge invariance by BRST symmetry. A quadratic infrared kernel for gauge and ghost fields is generally not BRST invariant because the BRST transformation is nonlinear. Introducing the usual BRST sources or antifields, the exact finite-scale statement has the schematic form

$$\mathcal S(\Gamma_k) = \Delta_k[\Gamma_k;R_k],$$

where $\mathcal S$ is the Slavnov functional and $\Delta_k$ is a regulator insertion containing the full regularized propagator. Its detailed signs and factors depend on ghost, source, field-ordering, and Legendre-transform conventions; those conventions must accompany an implementation.

Ellwanger derived such modified Slavnov–Taylor identities for Yang–Mills flow equations. The regulator terms supplement the standard identity, their evolution is compatible with the exact flow, and they vanish with the infrared kernels so that the standard identity is recovered at $k=0$. Ellwanger 1994, preprint pp. 2 and 5–6

For a truncation, “the breaking vanishes because $R_k\to0$” is not enough. One must evaluate

$$\varepsilon_{\mathrm{mSTI}}(k) = \left\| \mathcal S(\Gamma_{k,N}) - \Delta_k[\Gamma_{k,N};R_k] \right\|_{\mathcal P_{\mathrm{id}}},$$

with declared identity projectors $\mathcal P_{\mathrm{id}}$. The residual must be monitored in all tensor and momentum channels used by the physical claim. Tuning one relevant coupling to cancel one projected component does not establish the full identity.

The practical closure problem is coupled:

$$\begin{cases} \partial_t\Gamma_{k,N} =P_N\mathcal F_k[\Gamma_{k,N}],\\[2mm] \mathcal P_{\mathrm{id}} \left[ \mathcal S(\Gamma_{k,N})-\Delta_k \right] \approx0. \end{cases}$$

If ansatz enlargement improves the flow observable but worsens the identity residual, the larger truncation has not yet produced a stronger gauge-theory result.

Background symmetry is not fluctuation symmetry

In a background-field calculation, write a total field as background plus fluctuation. A regulator constructed covariantly from the background can preserve background gauge transformations, which is computationally useful. It nevertheless introduces separate background and fluctuation dependence.

For a linear split, the corresponding splitting identity has the schematic structure

$$\frac{\delta\Gamma_k}{\delta\bar\phi} - \frac{\delta\Gamma_k}{\delta\phi} = \frac12\operatorname{STr} \left[ G_k\frac{\delta R_k[\bar\phi]}{\delta\bar\phi} \right] + \text{gauge-fixing and ghost terms}.$$

The precise identity changes for nonlinear splits and field-space measures. Safari derives the general modified splitting Ward identity and shows how its infrared limit constrains the effective action’s background dependence. Safari 2016, preprint § 2.1, pp. 3–5, and § 7, pp. 20–21

Thus the statements

1. $\Gamma_k$ is invariant under a simultaneous background transformation,
2. the fluctuation vertices satisfy the physical Slavnov–Taylor identity, and
3. the result is independent of the auxiliary background split

are not interchangeable. A single-field approximation $\Gamma_k[\bar\phi,\phi]\mapsto\Gamma_k[\bar\phi+\phi]$ imposes the third relation rather than deriving it; its error must be checked with a split-identity residual or a bimetric enlargement.

Convexity, poles, and the infrared endpoint

For ordinary bosonic Legendre directions, the endpoint effective action is convex. At finite $k$, the path integral instead guarantees positivity of the regulated inverse propagator on its domain:

$$\Gamma_k^{(2)}+R_k>0, \qquad g_k \equiv \inf\operatorname{spec} \left(\Gamma_k^{(2)}+R_k\right)>0.$$

The unregulated Hessian $\Gamma_k^{(2)}$ may still have negative eigenvalues in a symmetry-breaking region. As $k\to0$, the flow can flatten the inner part of a scalar effective potential while $g_k$ approaches zero from above. A solver that steps through $g_k=0$ has crossed a pole of the defining equation; smaller step size does not turn the continued branch into a solution.

Spectral analyses show how convexity follows from suitable functional flows and also why regulator and truncation conditions matter. In particular, expanding the propagator around an incomplete Hessian can preserve convexity only for the retained part rather than for the full truncated action. Litim, Pawlowski, and Vergara 2006, preprint pp. 2–4

Convexity is not a blanket positivity statement for gauge-fixed gauge fields, ghosts, or fermionic Hessians. There the relevant test is invertibility of the full graded regularized operator together with BRST or modified-identity control.

Redundant directions and field normalization

An infinitesimal field redefinition $\varphi\mapsto\varphi+\epsilon\Psi[\varphi]$ moves the action along

$$\mathcal R_\Psi\Gamma = \int d^dx\, \Psi[\varphi](x) \frac{\delta\Gamma}{\delta\varphi(x)}.$$

This is a redundant direction: it changes coordinates on theory space rather than an observable. A finite regulator and projection can make reparameterization invariance imperfect, so changing the field-normalization condition may move approximate fixed-point coordinates and even a spurious stability eigenvalue.

Report the normalization point, transform observables consistently, and test at least one alternative normalization. Do not count a redundant eigenvector as a new physical scaling operator merely because it appears in the finite stability matrix. The next chapter separates physical scaling directions from coordinate choices.

Minimum FRG validation checklist

Use this same record for scalar, fermionic, gauge, and background-field flows, marking a check inapplicable only with a physical reason.

Record	Declare before solving	Required check	Failure that blocks the claim
Regulator and endpoints	Kernel for every field species, shape parameters, normalization, ultraviolet data, and infrared removal limit	Repeat with admissible regulator choices and verify both endpoint conditions	A singular trace, unmatched endpoints, or an observable that moves beyond the stated range
Ansatz and omitted structures	Fields, operators, derivative order, vertex order, field domain, and every deliberately omitted channel	Enlarge the ansatz in at least one physically relevant direction	No explicit account of what the next enlargement adds
Projection and coordinates	Field values, external momenta, tensor normalization, running basis, and expansion point	Change projection point or representation and include all chain-rule terms	An unsupported dependence on projector kinematics or a singular coordinate chart
Closure	Treatment of higher vertices, operators, and momentum dependence requested by the flow	Compare a distinct closure or bound the discarded functional residual	A hidden replacement of an omitted structure by zero or by classical data
Nested truncations	At least three successive orders when available, with identical inputs and observables	Report signed values and successive differences rather than only the final order	A digit retained beyond the largest relevant nested change
Symmetry identity	Exact, modified, or broken identity appropriate to the regulator and approximation	Evaluate the identity residual independently of the projected flow equations	A residual comparable to the retained physical signal without a qualified claim
Convexity and invertibility	Domain on which $\Gamma_k^{(2)}+R_k$ must be invertible and the expected infrared convexity behavior	Monitor its smallest relevant eigenvalue and field-domain dependence	An unhandled Hessian pole, unstable branch, or claimed endpoint before convexification
Independent benchmark	Analytic limit, perturbative coefficient, solvable model, alternative method, or external data	Reproduce the benchmark without tuning inputs to its answer	Unmatched inputs, circular calibration, or unexplained discrepancy
Numerics and justified digits	Solver, tolerances, grid or basis, convergence criterion, precision, and uncertainty prescription	Vary resolution and tolerance and retain only digits stable under all larger effects	Solver convergence alone or more printed digits than the uncertainty supports

A decomposed error budget

For an observable $O$, keep a vector of diagnostics rather than immediately compressing everything into one number:

$$\boldsymbol\delta_O = \left( \delta_{\mathrm{num}}, \delta_{\mathrm{reg}}, \delta_{\mathrm{ansatz}}, \delta_{\mathrm{proj}}, \delta_{\mathrm{closure}}, \delta_{\mathrm{sym}}, \delta_{\mathrm{UV/IR}}, \delta_{\mathrm{bench}} \right).$$

Component	O(N) scalar implementation	Gauge or background-field implementation
Numerical	Grid, basis, step, precision, and field-domain variation	Add momentum routing, tensor projection, and linear-solver variation
Regulator	Optimized versus smooth exponential profiles with matched normalization	Vary admissible gauge/ghost kernels without changing the physical inputs
Ansatz and closure	Raise polynomial and derivative order; compare polynomial with field grid	Add vertices and tensor structures required by the mSTI or split identity
Projection	Change field point and momentum configuration	Change transverse/longitudinal and identity projectors coherently
Symmetry	Monitor $\mathcal E_W$ on the full field domain	Monitor projected mSTI and split-identity residuals along the entire flow
Endpoint	Check $R_k\to0$, $g_k>0$, and convexification	Check regulator removal, graded invertibility, and restoration of the physical identity
Benchmark	Recover the universal one-loop coefficient and a solvable limit	Recover perturbative Ward/Slavnov–Taylor relations or an independent gauge-invariant observable

These effects are correlated. Regulator dependence, for example, is generated by the truncation and should not be added as though it were statistically independent of ansatz error. Give the component table first. If one scalar uncertainty is required, state a conservative rule such as the largest accepted excursion or a linear sum of independently bounded components.

For the $O(N)$ application, the one-loop coefficient and Ward residual are pass/fail checks, not uncertainty contributions. At a nonperturbative fixed point, quote a physical exponent only after regulator, derivative-order, potential-representation, projection-point, and numerical variations have all been performed on matched inputs.

Reliability criteria for physical claims

Stop or narrow the claim when any of the following occurs:

• the finite-$k$ modified identity cannot be represented by the ansatz;
• the identity residual is comparable to the physical tensor or vertex being quoted;
• the standard identity is not restored as the regulator is removed;
• $\Gamma_k^{(2)}+R_k$ develops an unhandled zero mode;
• nested ansätze or distinct projections move the observable beyond the proposed range;
• a regulator plateau disappears at the next truncation order;
• a field-normalization change moves a supposedly physical result without the corresponding coordinate transformation;
• a benchmark fails on matched inputs; or
• printed digits exceed the largest accepted diagnostic variation.

Passing every internal check supports a statement about stability within the explored family. It is not a proof that all omitted operators are small. Stronger claims require an independent analytic limit, perturbation theory, simulation, experiment, or a distinct nonperturbative method.

Common pitfalls

Testing the wrong identity. If the regulator breaks a symmetry, the unmodified identity is not expected at finite $k$. Compare against the derived regulator insertion and test the unmodified identity only at the physical endpoint.

Equating background invariance with gauge independence. A background-covariant regulator can preserve simultaneous transformations while fluctuation vertices still satisfy modified identities and retain split dependence.

Using an invariant ansatz as a convergence test. Structural $O(N)$ invariance can force a Ward residual to zero even in a very small ansatz. It checks symmetry implementation, not completeness.

Continuing through a Hessian pole. An apparently smooth numerical branch beyond a zero of the regulated inverse propagator is not validated by solver convergence.

Exercises

1. Derive the modified linear Ward identity from the change of variables $\chi\mapsto\chi+\epsilon T\chi$.

Solution

Invariance of the microscopic action and measure gives

$$J\cdot T\varphi = \left\langle\delta\Delta S_k/\epsilon\right\rangle = \frac12\varphi\cdot K_{T,k}\cdot\varphi + \frac12\operatorname{Tr}(K_{T,k}G_k).$$

The modified Legendre transform implies $\Gamma_k^{(1)}=J-R_k\varphi$. Because $(R_k\varphi)\cdot T\varphi=\varphi\cdot K_{T,k}\cdot\varphi/2$, subtracting it cancels the mean-field term and leaves $\Gamma_k^{(1)}\cdot T\varphi=\operatorname{Tr}(K_{T,k}G_k)/2$.

2. Verify the profile-independent threshold integral for $R_k=q^2r(q^2/k^2)$.

Solution

At fixed $q$, $\partial_tR_k=-2q^2y r'(y)$. In four dimensions,

$$\int_q f(q^2) = \frac{k^4}{16\pi^2} \int_0^\infty dy\,y\,f(k^2y).$$

Substitution gives

$$\mathcal I_R = - \frac1{8\pi^2} \int_0^\infty dy\, \frac{r'(y)}{(1+r(y))^3} = \frac1{16\pi^2} \left[\frac1{(1+r)^2}\right]_0^\infty.$$

The declared endpoint conditions make the bracket equal to one. A profile that violates those conditions need not reproduce the universal coefficient.

3. Why can a background-gauge-invariant truncation still give an incorrect physical gluon vertex?

Solution

Background invariance constrains simultaneous transformations of background and fluctuation fields. The physical fluctuation vertices also obey a modified Slavnov–Taylor identity, while equality between background and fluctuation dependence is controlled by a splitting identity. A truncation can satisfy the first constraint by construction while violating either of the other two, so all relevant residuals must be checked independently.

The chapter has now supplied the exact Wilsonian and effective-average-action flows, finite projections, and their validation conditions. Fixed Points and Universality uses that machinery to classify scaling solutions without confusing physical eigenoperators with redundant coordinates.

References

• De Polsi, Gonzalo, and Nicolás Wschebor. “Regulator Dependence in the Functional Renormalization Group: A Quantitative Explanation.” Physical Review E 106 (2022) 024111. DOI. Open PDF
• Ellwanger, Ulrich. “Flow Equations and BRS Invariance for Yang–Mills Theories.” Physics Letters B 335 (1994): 364–370. DOI. Open PDF
• Knorr, Benjamin. “Exact Solutions and Residual Regulator Dependence in Functional Renormalisation Group Flows.” Journal of Physics A: Mathematical and Theoretical 54 (2021) 275401. DOI. Open PDF
• Litim, Daniel F., Jan M. Pawlowski, and Lautaro Vergara. “Convexity of the Effective Action from Functional Flows.” Preprint (2006). arXiv:hep-th/0602140. Open PDF
• Safari, Mahmoud. “Splitting Ward Identity.” Preprint (2016). arXiv:1508.06244. Open PDF